Jurnal Ficks law.pdf

8/10/2019 Jurnal Ficks law.pdf

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The Open Environmental EngineeringJournal, 2011, 4, 105-111 105

1874-8295/11 2011 Bentham Open

Open Access

Modeling Diffusive Flux of Non Point Source Pollutants in Lake Victoria:A Comparison Study of Ficks Law and the Fokker-Planck Law

N. Banaddaa,

*, F. Ayaaa, I. Nhapi

b, U.G. Wali

c, R.J. Kimwaga

dand D.A. Mashauri

d

aCollege of Agricultural and Environmental Sciences, Makerere University, P.O. Box 7062, Kampala, Uganda

bDepartment of Civil Engineering, University of Zimbabwe, Box MP 167, Harare, Zimbabwe

cFaculty of Applied Sciences, National University of Rwanda, P.O. Box 117, Butare, Rwanda

dCollege of Engineering, University of Dar es Salaam, P. O. Box 35131, Dar es Salaam, Tanzania

Abstract:Mathematical models have the potential to conceptually quantify, link and simulate the interactive processes of

nature. In this study 68 samples were collected at Gaba landing site in Uganda during a rainy season and were analyzed

for nutrients, namely, Ammonia, Nitrite, Nitrate, and Phosphate. In addition, portable meters were used to measure Total

Dissolved Solids (TDS) and Dissolved Oxygen (DO) instantaneously at point of sample collection. Within the lake, sam-

ples were taken at for horizontal transects of 10 metres (m) interval over a distance of 50 m from the shore where surface

runoff was released. At each 10 m sampling point, three samples were drawn at vertical distances of 0.5 m, 1.0 m and 1.5

m from water surface using a hand pump with graduated delivery pipe. This paper presents the results obtained from theapplication of two alternative expressions, ficks law and Fokker-Planck law to gain insight into the pollutants diffusive

flux patterns within the lake. We conclude that in general the Fokker-Planck model should be given preference, in model-

ling Ammonia and Phosphate flux profiles while Fickian model should be deployed in modelling DO, TDS, Nitrites and

Nitrates.

Keywords:Modeling and identification, control applications in environmental processes, water quality, nutrients,lake Victoria.

1. INTRODUCTION

Diffusion is one of the most fundamental phenomena innature.Studies on diffusion processes in rivers and lakes arewidely used by hydrodynamicists, hydrologists and envi-ronmental scientists involved in the study of water pollutionproblems. The time of travel of a pollutant in a river or lake,the rate at which the pollutant spreads, the decrease in peakconcentration and the resulting concentration patterns ofpollutant are the important variables that must be properlyunderstood. Serious pollution may result if the capacity ofthe lake or steam to transport and disperse a contaminant isoverestimated. Underestimation on the other hand may resultin valuable resources not being optimally utilized, resultingin unnecessary expenditure on treatment facilities. The Na-tional Water and Sewerage Corporation charged with treat-ing and supplying water to Kampala City dwellers are com-plaining of raising costs for treating Lake Victoria water [1].A number of studies e.g., [2-6] have documented pollution as

one of the major problems causing water quality deteriora-tion. This is unfortunate due to the socio-economic impor-tance of the lake. Lake Victoria is not only a source of food(fish), water, employment, transport, hydroelectric power,and recreation. In spite of the enormous attention that hasbeen invested in tackling the pollution problem during the

*Address correspondence to this author at the College of Agricultural andEnvironmental Sciences, Makerere University, P.O. Box 7062, Kampala,Uganda; Tel & Fax: +256-41-53.16.41; E-mail: [email protected]

last three decades, it has persisted. In this study, Non- poinsource pollution (NPS) is mainly been the focus. Globally30 to 50% of the earths surface is believed to be affected byNPS pollutants [7-9]. Agriculture remains as the singlegreatest contributor of these pollutants to soil and water re

sources [10]. The most common NPS pollutants includeeroded sediments, fertilizers, pesticides, organic manuressalts, trace elements and sewage sludge. As pointed out by[11], the magnitude of NPS pollution is so complex and difficult to characterize that it defies proper evaluation withrespect to its environmental impact. Nevertheless, NPS pollutants are recognized as the major contributors to surfaceand groundwater contamination worldwide [12]. Even withinthe highly industrialized, NPS pollutants impair far morerivers and lakes than point sources [13]. There have beenmany studies on dispersion and diffusion of pollutants inrivers and lakes. Probably the best-known study, usuallyquoted in texts on the subject, is the Fickian theory. It wadeveloped for flow in pipes by [14, 15] and extended to

channels by [16] and for natural streams by [17]. The proliferation of GIS-based models of NPS pollutants can be causefor optimism and caution [18]. On the positive side, visualization with GIS provides decision makers with a means oquickly digesting complex interrelated spatial patters of NPSdistributions. On the negative side, unless these visualizations are accompanied with associated uncertainties, they canbe falsely regarded as black and white representations of thetruth, thereby creating the illusion of legitimacy. Againssuch a backdrop, this study will test the apparent popularity


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106 The Open Environmental EngineeringJournal, 2011, Vol. 4 Banadda et al.

of Fickian-type model by comparing it with the Fokker-Planck model. The global aim of this study is to comparetwo alternative expressions, ficks law and Fokker-Plancklaw to gain insight into the pollutants, namely, ammonia,nitrite, nitrate, phosphate, Dissolved Oxygen and Total Sus-pended Solids diffusive flux patterns within the lake.

2. MATERIALS AND METHODS

2.1. Study Area

The study was carried out at Gaba landing site located in

Makindye division, Kampala (Uganda) as shown in Fig. (1).The choice of the study area was informed by it being a highactivity area as far as NPS pollution is concerned.

2.2. Sample Collection and Analysis

In this study 68 samples were collected at Gaba landingsite in Uganda during a rainy season. To ensure that sampleswere always taken from the same spot within the lake, map-ping of the sampling points were done using GPS on a boatride. The sampling coordinates were stored in a GPS andlater traced during subsequent sampling. The mapping wasdone after rain event in order to locate the path/areas theserunoff normally follow when released into the Lake. Sampleswere stored in a cooler and transported within less than twohours for lab analysis which include nutrients (Ammonia,Nitrite, Nitrate, and Phosphate). Total Dissolved Solids(TDS) and Dissolved Oxygen (DO) were measured instanta-neously at point of sample collection. Meanwhile within thelake samples were taken at for horizontal transects of 10minterval over a distance of 50m starting from the shore wheresurface runoff was released. For the same sampling pointswithin the lake, samples were drawn at vertical distances of0.5m, 1.0m and 1.5m from water surface using a hand pumpwith graduated delivery pipe so as to take samples at therequired vertical depth.

2.3. Model Selection and Modeling Approach

The modeling of (particle) diffusion is normally based ontwo assumptions. The first is the conservation of the numbeof particles:

(1

where n(x, t) is the particle density and (x, t) is the particleflux through the pointx. The second assumption is a relationbetween the flux, , and the density. If one has a full theoryfor the particle motion (kinetic or microscopic), one mayderive directly from the particle dynamics which is FickLaw as shown in Equation (2). However, if a theory is lacking or incomplete, it is customary to apply a phenomenological relation.

(2

where D is the diffusion coefficient. For Fick law, we assumed a constant D of 10-4[19]. To reconcile the experimentally observed fluxes with theory, often ad hoc convective

terms (i.e., V (x) n) or drifts are added to the particle fluxwhich in many cases are difficult to justify physically [20]However, the need for these drifts and their physical interpretation can be perfectly justified from a theoretical standpoint without such additional terms. Indeed, it has long beenknown (although widely ignored) that the diffusion is subjecto a spatial inhomogeneity satisfies the so-called FokkerPlanck diffusivity law as depicted in Equation (3).

(3

Fig. (1).Land Use Activities in Gaba, Kampala (Uganda).


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Modeling Diffusive Flux of Non Point Source Pollutants in Lake Victoria The Open Environmental EngineeringJournal, 2011, Volume 4 107

Where D = 0.0142E1.49

with E being the depth (m) ac-cording to [21]. Diffusion is a macroscopic process that re-flects underlying random motion at a microscopic level. Themain result of this formalism is thegeneralized master equa-tion, which provides a probabilistic description of the micro-scopic motion. In the case of Markovian particle motion (inwhich each random step of every particle is taken from aprobability distributionp without any influence from the pasthistory of the system) as shown in Equation (4).

(4)

In spite of the apparent complexity of this integro-differential equation, its interpretation is easy. It states that

the rate of change of particles at location x and time t isgiven by adding the new particles that arrive and subtractingthose that leave, per time unit. Those arriving move in fromanother location x by a process with a probability given byp(xx, x , t) . The time scale governing the motion is givenby D(x).

3. RESULTS AND DISCUSSIONS

3.1. Dissolved Oxygen (DO) Flux Profiles

Fig. (2) depicts the dissolved Oxygen flux profiles mod-elled by Fick law and Fokker-Planck law. Generally, Ficks

model shows a decreasing trend in DO flux values as depthincreases from 0.5 metres to 1.5 metres. On the other handFokker-Planck model shows an increasing trend in DO fluxover a similar depth. Measured DO data showed a decreasingDO trend with increasing depth [22, 23]. At a depth of 0.5 mhigher DO flux values were obtained at the shores and subsequently decreased with increasing horizontal distance. Thiis due to inflow of polluted runoff. A comparison of the twomodels shows that Fickian model is closer to reality as compared to Fokker- Planck model. At a horizontal distance oapproximately 25 m, depth of 1.0 m and 1.5 m, a markedincrease in DO flux is observed.

3.2. Total Dissolved Solids (TDS) Flux Profiles

Fig. (3) depicts the Total Dissolved Solids (TDS) dis

solved Oxygen flux profiles modelled by Fick law and Fok

ker-Planck law. It can be noted that Fickian model shows a

decrease in TDS flux values with increase in depth, although

similar values were obtained at depth of 1 m and 1.5 m. In

terestingly, Fokker-Planck model show increasing TDS flux

values with increase in depth. Field TDS measurement

showed a decreasing pattern in TDS values with increase in

depth. At the shore, higher TDS flux is noted and decrease a

one moves horizontally into the lake. A comparison of the

two models shows that Fickian model is closer to reality a

compared to Fokker- Planck model.

Fig. (2).Dissolved Oxygen flux profiles at a depth of (a) 0.5 m (b) 1.0 m and (c) 1.5 m.


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3.3. Ammonia Flux Profiles

Fig. (4) shows the Ammonia flux profiles modelled byFick law and Fokker-Planck law. Fickian model showshigher Ammonia flux values at the shore and a depth of 0.5

m. At the shore but a depth of 1.5 m, zero flux is noted. Thismeans that there is no concentration gradient as runoff enterthe lake. Fickian model Ammonia Flux values stagnant aapproximately 1.3 mgL

-1m

-2s

-1 with increased depth and

Fig. (3).Total Dissolved Solids (TDS) Flux Profiles at a depth of (a) 0.5 m (b) 1.0 m and (c) 1.5 m.

Fig. (4).Ammonia Flux Profiles at a depth of (a) 0.5 m (b) 1.0 m and (c) 1.5 m.


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horizontal distance. On the other hand, Fokker-Planck modelvalues increase with increasing depth and horizontal dis-tance. Fickian model in Fig. (1) seemed to agree with litera-ture e.g., [23] as far as DO is concerned. Therefore, anoxia

seems a reality. Anoxia in the bottom waters is producedthrough lack of resupply of oxygen from the surface andbacteria consuming the oxygen through respiration. Bacterialbreakdown of dead plant and animal material producesammonium as a by-product. Due to limited mixing,ammonium concentrations increase and become quite high inthe bottom waters. In this regard, Fokker-Planck modelseems closer to measured Ammonia values.

3.4. Nitrites and Nitrates Flux Profile

In Fig. (4and 5), it can be noted that Nitrite and Nitratefluxes are higher at the shore and depth of 0.5 m. This is dueto surface runoff entering the lake. Generally, Fickian modelNitrite and Nitrate values decreased with increased depth and

horizontal distance. However, at all the three depths consid-ered in this study, Nitrate flux was higher than Nitrite flux.Both models reported this trend and increase in Nitrate fluxas compared to Nitrite. This concurs with what is reported inliterature as [8] in their works noted that Nitrite concentratesshowed an inverse relationship to Nitrate concentrates. Fok-ker-Plnack model indicate that as Nitrate flux value increaseso do Nitrites. Fickian model indicate that as Nitrate fluxvalues increase, Nitrite flux values decrease. In the surfacewater, DO concentration is high enough to oxidize Nitrites toNitrates. Therefore, under these circumstances, Fickianmodel is closer to reality.

3.5. Phosphates Flux Profile

In Fig. (6), Fokker-Planck model shows that the phos

phates flux increase with horizontally at depth of 1 m and 1.5

m. On the hand, the Fickian model depicts an increase in

phosphate flux with increase in horizontal distance at depth

of 1 m and 1.5 m. However, the Fickian model shows tha

phosphate flux decrease with increase in depth while the

Fokker-Planck model depicts an inverse trend. Literatur

supports the trend observed in the Fokker-Planck model. A

study carried out by [23] shows that phosphate concentra

tions are high with increasing depth.

4. CONCLUSIONS & PERSPECTIVES

This work presents the results of application of two alter

native expressions, ficks law and Fokker-Planck law to gain

insight into the pollutants diffusive flux patterns within Lak

Victoria. Mathematical modelling appears to be a powerfutool to predict pollution diffusive flux in the case study pre-

sented. ON one hand, Fickian model showed a satisfactory

agreement with measured data in predicting DO, TDS, Ni

trites and Nitrates flux profiles. On the other hand, Fokker

Planck model should be given preference in modeling

Phopshates and Ammonia flux profiles. However, the Fick

ian model is more popular and largely deployed than the

Fokker-Planck model. Future works should focus on disper

sion studies, i.e., variations of NPS pollutant concentrations

with time and space. In general, it can be concluded that:

Fig. (5).Nitrates Flux Profiles at a depth of (a) 0.5 m (b) 1.0 m and (c) 1.5 m.


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Increase in land use & urbanization in shore settlementsof Lake Victoria has increased nutrient and dissolvedsolids concentration in surface runoff resulting into dete-rioration in their quality

Surface runoff discharged into Lake Victoria causes an

elevation in nutrient and physico-chemical parameterconcentrations

Availability of fish especially in environs close to shoresettlements is diminished due to high water pollutionloads.

ACKNOWLEDGEMENTS

Acknowledgement is made to SIDA/SAREC through theInter University Council for Eastern Africa that funded thiswork under the Lake Victoria Research (VICRES) pro-gramme.

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Received: December 06, 2010 Revised: February 15, 2011 Accepted: February 22, 2011

Banaddaet al.; LicenseeBentham Open.This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License

(http://creativecommons.org/licenses/by-nc/3.0/g) which permits unrestricted, non-commercial use, distribution and reproduction in any medium, provided thwork is properly cited.

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